Averaging the Truth Value of Formulas in Gödel Logic

Let
$L$
be a propositional mathematical fuzzy logic with the real unit interval [0, 1] as its set of truth values. Assume that
$L$
has an algebraic semantics given by a variety V of algebras. A finitely additive probability measure (or, state) over the free n-generated V-algebra provides an average value over all assignments of a formula with
$n$
many variables in
$L$
only if this measure is invariant with respect to automorphisms of the free algebra. In this paper we characterise the states that are invariant with respect to automorphisms of the free n-generated Gödel algebra.